Title:Exponentially small asymptotic estimates for the splitting of separatrices to whiskered tori with quadratic and cubic frequencies

Abstract: We study the splitting of invariant manifolds of whiskered tori with two or
three frequencies in nearly-integrable Hamiltonian systems. We consider
2-dimensional tori with a frequency vector $\omega=(1,\Omega)$ where $\Omega$
is a quadratic irrational number, or 3-dimensional tori with a frequency vector
$\omega=(1,\Omega,\Omega^2)$ where $\Omega$ is a cubic irrational number.
Applying the Poincare-Melnikov method, we find exponentially small asymptotic
estimates for the maximal splitting distance between the stable and unstable
manifolds associated to the invariant torus, showing that such estimates depend
strongly on the arithmetic properties of the frequencies. Inthe quadratic case,
we use the continued fractions theory to establish a certain arithmetic
property, fulfilled in 24 cases, which allows us to provide asymptotic
estimates in a simple way. In the cubic case, we focus our attention to the
case in which $\Omega$ is the so-called cubic golden number (the real root of
$x^3+x-1=0$), obtaining also asymptotic estimates. We point out the similitudes
and differences between the results obtained for both the quadratic and cubic
cases.